Some energy-related functionals and their vertical variational theory
The energy and volume of a mapping of Riemannian manifolds are linked by a discrete family of functionals, indexed by the elementary symmetric polynomials. We explore the variational properties of members of this family; in particular, their tension fields, stress-energy tensors, and Jacobi operators. When one Riemannian manifold fibres over another, applying the conventional theory of harmonic maps to sections neglects the additional structure supplied by the fibering. We give an alternative definition for harmonicity of sections which overcomes this deficiency, and is closely- enough linked to the conventional theory to share many of the qualitative properties of harmonic maps. Such "harmonic sections" arise as solutions to a variational problem, one consequence of which is to allow the proof of a reduction theorem for harmonic sections of a Riemannian vector bundle. The "Gauss section" of an isometrically immersed submanifold extends the idea of "Gauss map" to ambient spaces of arbitrary curvature. We prove a non-trivial identity relating Gauss map to second fundamental form, and generalize to the Gauss section. This leads to a characterization of immersions with harmonic Gauss section, and to a further identity involving a suitably generalized notion of "third fundamental" form". We are also able to characterize those Riemannian foliations of codinension one whose Gauss section is harmonic.